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Theorem disjss2 4075
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )

Proof of Theorem disjss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3250 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2694 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 3040 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A
y  e.  B ) )
42, 3syl 15 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A y  e.  B
) )
54alimdv 1621 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( A. y E* x  e.  A
y  e.  C  ->  A. y E* x  e.  A y  e.  B
) )
6 df-disj 4073 . 2  |-  (Disj  x  e.  A C  <->  A. y E* x  e.  A
y  e.  C )
7 df-disj 4073 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x  e.  A
y  e.  B )
85, 6, 73imtr4g 261 1  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1540    e. wcel 1710   A.wral 2619   E*wrmo 2622    C_ wss 3228  Disj wdisj 4072
This theorem is referenced by:  disjeq2  4076  0disj  4095  uniioombllem2  19036  uniioombllem4  19039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-ral 2624  df-rmo 2627  df-in 3235  df-ss 3242  df-disj 4073
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